Abstract

We describe the effects of neutrino propagation in the matter of the Earth relevant to experiments with atmospheric and accelerator neutrinos and aimed at the determination of the neutrino mass hierarchy and CP violation. These include (i) the resonance enhancement of neutrino oscillations in matter with constant or nearly constant density, (ii) adiabatic conversion in matter with slowly changing density, (iii) parametric enhancement of oscillations in a multilayer medium, and (iv) oscillations in thin layers of matter. We present the results of semianalytic descriptions of flavor transitions for the cases of small density perturbations, in the limit of large densities and for small density widths. Neutrino oscillograms of the Earth and their structure after determination of the 1–3 mixing are described. A possibility to identify the neutrino mass hierarchy with the atmospheric neutrinos and multimegaton scale detectors having low energy thresholds is explored. The potential of future accelerator experiments to establish the hierarchy is outlined.

1. Introduction

Neutrinos are eternal travelers: once produced (especially at low energies) they have little chance to interact and be absorbed. Properties of neutrino fluxes are flavor compositions, lepton charge asymmetries, and energy spectra of encode information. Detection of the neutrinos brings unique knowledge about their sources, properties of medium, the space-time they propagated as well as about neutrinos themselves.

Neutrino propagation in matter is vast area of research which covers a variety of different aspects: from conceptual ones to applications. This includes propagation in matter (media) with (i) different properties (unpolarized, polarized, moving, turbulent, fluctuating, with neutrino components, etc.), (ii) different density profiles, and (iii) in different energy regions. The applications cover neutrino propagation in matter of the Earth and the Sun, supernova and relativistic jets as well as neutrinos in the early universe.

The impact of matter on neutrino oscillations was first studied by Wolfenstein in 1978 [1]. He marked that matter suppresses oscillations of the solar neutrinos propagating in the Sun and supernova neutrinos inside a star. He considered hypothetical experiments with neutrinos propagating through 1000 km of rock, something that today is no longer only a thought but actual experimental reality. Later Barger et al. [2] have observed that matter can also enhance oscillations at certain energies. The work of Wolfenstein was expanded upon in papers by Mikheev and Smirnov [3–5], in particular, in the context of the solar neutrino problem. Essentially two new effects have been proposed: the resonant enhancement of neutrino oscillations in matter with constant and nearly constant density and the adiabatic flavor conversion in matter with slowly changing density. It was marked that the first effect can be realized for neutrinos crossing the matter of the Earth. The second one can take place in propagation of solar neutrinos from the dense solar core via the resonance region inside the Sun to the surface with negligible density. This adiabatic flavor transformation, called later the MSW effect, was proposed as a solution of the solar neutrino problem.

Since the appearance of these seminal papers, neutrino flavor evolution in background matter was studied extensively including the treatment of propagation in media which are not consisting simply of matter at rest, but also backgrounds that take on a more general form. For instance, in a thermal field theory approach [6], effects of finite temperature and density can be taken readily into account. If neutrinos are dense enough, new type of effects can arise due to the neutrino background itself, causing a collective behavior in the flavor evolution. This type of effect could have a significant impact on neutrinos in the early universe and in central parts of collapsing stars.

There has been a great progress in treatments of neutrino conversion in matter, both from an analytical and a pure computational points of view. From the analytical side, the description of three-flavor neutrino oscillations in matter is given by a plethora of formulas containing information that may be hard to get a proper grasp of without introducing approximations. Luckily, given the parameter values inferred from experiments, various perturbation theories and series expansions in small parameters can be developed. In this paper we will explain the basic physical effects important for the current and next generation neutrino oscillation experiments and provide the relevant formalism. We present an updated picture of oscillations and conversion given the current knowledge on the neutrino oscillation parameters.

In this paper we focus mainly on aspects related to future experiments with atmospheric and accelerator neutrinos. The main goals of these experiments are to (i) establish the neutrino mass hierarchy, (ii) discover CP violation in the lepton sector and determination of the CP-violating phase, (iii) precisely measure the neutrino parameters, in particular, the deviation of 2-3 mixing from maximal, and (iv) search for sterile neutrinos and new neutrino interactions.

Accelerator and atmospheric neutrinos propagate in the matter of the Earth. Therefore we mainly concentrate on effects of neutrino propagation in the Earth, that is, in usual electrically neutral and nonrelativistic matter. We update existing results on effects of neutrino propagation in view of the recent determination of the 1–3 mixing.

The paper is organized as follows. In Section 2 we consider properties of neutrinos in matter, in particular, mixing in matter and effective masses (eigenvalues of the Hamiltonian); we derive equations which describe the propagation. Section 3 is devoted to various effects relevant to neutrino propagating in the Earth. We consider the properties of the oscillation/conversion probabilities in different channels. In Section 4 we explore the effects of the neutrino mass hierarchy and CP-violating phase on the atmospheric neutrino fluxes and neutrino beams from accelerators. Conclusions and outlook are presented in Section 5.

2. Neutrino Properties in Matter

We will consider the system of 3-flavor neutrinos, , mixed in vacuum:
Here is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [7–9] and is the vector of mass eigenstates with masses (). We will use the standard parameterization of the PMNS matrix,
which is the most suitable for describing usual matter effects. In (2) are the matrices of rotations in the -planes with angles and .

In vacuum the flavor evolution of these neutrinos is described by the Schrödinger-like equation
where is the neutrino mass matrix in the flavor basis and is the neutrino energy. Equation (3) is essentially a generalization of the equation for a single ultrarelativistic particle. According to (3), the Hamiltonian in vacuum can be written as
where and we take the masses to be real (the term is omitted in (4) since it does not produce a phase difference).

2.1. Refraction and Matter Potentials

The effective potential for a neutrino in medium can be computed as a forward scattering matrix element . Here is the wave function of the system of neutrino and medium, and is the Hamiltonian of interactions.

At low energies, the Hamiltonian is the effective four-fermion Hamiltonian due to exchange of the and bosons:
where and are the vector and axial vector coupling constants.

In the Standard Model the matrix of the potentials in the flavor basis is diagonal: .

For medium the matrix elements of vectorial components of vector current are proportional to velocity of particles of medium. The matrix elements of the axial vector current are proportional to spin vector. Therefore for nonrelativistic and unpolarized medium (as well as for an isotropic distribution of ultrarelativistic electrons) only the component of the vector current gives a nonzero result, which is proportional to the number density of the corresponding particles. Furthermore, due to conservation of the vector current (CVC), the couplings and can be computed using the neutral current couplings of quarks. Thus, taking into account that, in the Standard Model, the neutral current couplings of electrons and protons are equal and of opposite sign, their NC contributionscancel in electrically neutral medium. As a result, the potential for neutrino flavor is
where and are the densities of electrons and neutrons, respectively.

Only the difference of potentials has a physical meaning. Contribution of the neutral current scattering to is the same for all active neutrinos. Since (, or a combination thereof) is due to the neutral current scattering, in a normal medium composed of protons neutrons (nuclei) and electrons, . Furthermore, the difference of the potentials for and is due to the charged current scattering of on electrons () [1]:

The difference of potentials leads to the appearance of an additional phase difference in the neutrino system: . This determines the refraction length, the distance over which an additional “matter” phase equals :
Numerically,
where is the nucleon mass. The corresponding column density is given by the Fermi coupling constant only.

For antineutrinos the potential has an opposite sign. Being zero in the lowest order the difference of potentials in the system appears at a level of due to the radiative corrections [10]. Thus in the flavor basis in the lowest order in EW interactions the effect of medium on neutrinos is described by with given in (7).

The potential has been computed for neutrinos in different types of media, such as polarized or heavily degenerate electrons, in [11–13].

2.2. Evolution Equation, Effective Hamiltonian, and Mixing in Matter

2.2.1. Wolfenstein Equation

In the flavor basis, the Hamiltonian in matter can be obtained by adding the interaction term to the vacuum Hamiltonian in vacuum [1, 3–5, 14, 15]:
In (10) we have omitted irrelevant parts of the Hamiltonian proportional to the unit matrix. The Hamiltonian for antineutrinos can be obtained by the substitution

There are different derivations of the neutrino evolution equation in matter, in particular, strict derivations starting from the Dirac equation or derivation in the context of quantum field theory (see [16] and references therein).

Although the Hamiltonian describes evolution in time, with the connection , (12) can be rewritten as with , so it can be used as an evolution equation in space.

Due to the strong hierarchy of and the smallness of 1–3 mixing, the results can be qualitatively understood and in many cases quantitatively described by reducing evolution to evolution. The reason is that the third neutrino effectively decouples and its effect can be considered as a perturbation. Of course, there are genuine phenomena such as CP violation, but even in this case the dynamics of evolution can be reduced effectively to the dynamics of evolution of systems. The evolution equation for two-flavor states, , in matter is
where the Hamiltonian is written in symmetric form.

2.3. Mixing and Eigenstates in Matter

The mixing in matter is defined with respect to —the eigenstates of the Hamiltonian in matter .

As usual, the eigenstates are obtained from the equation
where are the eigenvalues of . If the density and therefore are constant, correspond to the eigenstates of propagation. Since , the states differ from the mass states, . For low density , the vacuum eigenstates are recovered: . If the density, and thus change during neutrino propagation, and should be considered as the eigenstates and eigenvalues of the instantaneous Hamiltonian: , , and . For we have .

The mixing in matter is a generalization of the mixing in vacuum (1). Recall that the mixing matrix in vacuum connects the flavor neutrinos, , and the massive neutrinos, . The latter are the eigenstates of Hamiltonian in vacuum: . Therefore, the mixing matrix in matter is defined as the matrix which relates the flavor states with the eigenstates of the Hamiltonian in matter :

From (13) we find that
Furthermore, the Hamiltonian can be represented in the flavor basis as
Inserting this expression as well as the relation , which follows from (14), into (15) one obtains
or in matrix form . Thus, the mixing matrix can be found diagonalizing the full Hamiltonian. The columns of the mixing matrix, , are the eigenstates of the Hamiltonian which correspond to the eigenvalues . Indeed, it follows from (17) that .

Equation (14) can be inverted to , or in components . According to this, the elements of mixing matrix determine the flavor content of the mass eigenstates so that gives the probability to find in a given eigenstate . Correspondingly, the elements of the PMNS matrix determine the flavor composition of the mass eigenstates in vacuum.

2.4. Mixing in the Two-Neutrino Case

In the case, there is single mixing angle in matter and the relations between the eigenstates in matter and the flavor states read
The angle is obtained by diagonalization of the Hamiltonian (12) (see previous section):
where is the resonance factor. In the limit , the factor and the vacuum mixing are recovered. The difference of eigenvalues equals
This difference is also called the level splitting or oscillation frequency, which determines the oscillation length: (see Section 3.2).

The matter potential and always enter the mixing angle and other dimensionless quantities in the combination
where is the refraction length. This is the origin of the “scaling” behavior of various characteristics of the flavor conversion probabilities. In terms of the mixing angle in matter the Hamiltonian can be rewritten in the following symmetric form:

2.4.1. Resonance and Level Crossing

According to (19) the effective mixing parameter in matter, , depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, . The dependence for two different values of the vacuum mixing angle, corresponding to angles from the full three-flavor framework, is shown in Figure 1. The dependence of on has a resonant character [3]. At
the mixing becomes maximal: (). The equality in (23) is called the resonance condition and it can be rewritten as . For small vacuum mixing the condition reads the following: oscillation length refraction length. The physical meaning of the resonance is that the eigenfrequency, which characterizes a system of mixed neutrinos, , coincides with the eigenfrequency of the medium, . The resonance condition (23) determines the resonance density
The width of resonance on the half of height (in the density scale) is given by . Similarly, for fixed one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as , where . When the mixing approaches its maximalvalue: , the resonance shifts to zero density: , and the width of the resonance increases converging to the fixed value: .

Figure 1: Resonance in neutrino mixing. The dependence of on the product for vacuum mixing: , eV2 (red) and , eV2 (green). The left semiplane corresponds to antineutrinos. The behavior of with vacuum value is included for completeness. The dashed lines are the predictions from a strict two-flavor approximation while the solid thin lines are the results of numerical diagonalization of the full three-flavor system. The upper panels show the case of the normal mass hierarchy and the lower panels show the inverted hierarchy.

In a medium with varying density, the layer in which the density changes in the interval is called the resonance layer. In this layer the angle varies in the interval from to .

For , the mixing angle is close to the vacuum angle: , while for , the angle becomes and the mixing is strongly suppressed. In the resonance region, the level splitting is minimal [17, 18], therefore the oscillation length, as the function of density, is maximal.

2.5. Mixing of 3 Neutrinos in Matter

To a large extent, knowledge of the eigenstates (mixing parameters) and eigenvalues of the instantaneous Hamiltonian in matter allows the determination of flavor evolution in most of the realistic situations (oscillations in matter of constant density, adiabatic conversion, and strong breaking of adiabaticity). The exact expressions for the eigenstates and eigenvalues [19, 20] are rather complicated and difficult to analyze. Therefore approximate expressions for the mixing angles and eigenvalues are usually used. They can be obtained performing an approximate diagonalization of which relies on the strong hierarchy of the mass squared differences:
Without changing physics, the factor in the mixing matrix can be eliminated by permuting it with and redefining the state . Therefore, in what follows, we use . Here we will describe the case of normal mass hierarchy: . Subtracting from the Hamiltonian the matrix proportional to the unit matrix , we obtain

2.5.1. Propagation Basis

The propagation basis, , which is most suitable for consideration of the neutrino oscillations in matter, is defined through the relation
Since the potential matrix is invariant under 2-3 rotations, the matrix of the potentials is unchanged and the the Hamiltonian in the propagation basis becomes
It does not depend on the 2-3 mixing or CP violation phase, and so the dynamics of the flavor evolution do not depend on and . These parameters appear in the final amplitudes when projecting the flavor states onto propagation-basis states and back onto (27) the neutrino production and detection.

2.5.2. Mixing Angles in Matter

The Hamiltonian in (29) can be diagonalized performing several consecutive rotations which correspond to developing the perturbation theory in . After a 1–3 rotation
over the angle determined by
the 1–3 element of (29) vanishes. The expression (31) differs from that for mixing in matter by a factor , which increases the potential and deviates from 1 by
After this rotation the Hamiltonian in the basis (30) becomes

where
and . For , these elements are reduced to the standard expressions. In the limit of zero density, , , and consequently the 11-element of the Hamiltonian equals .

In the lowest approximation one can neglect the nonzero 2-3 element in (33). The state then decouples and the problem is reduced to a two-neutrino problem for . The eigenvalue of this decoupled state equals
The diagonalization of the remaining 1-2 submatrix is given by rotation
where is determined by
Here and are defined in (34) and (31), respectively. The eigenvalues equal

According to this diagonalization procedure in the lowest order in the mixing matrix in matter is given by
where mixing angles and are determined in (37) and (31), respectively. The 2-3 angle and the CP violation phase are not modified by matter in this approximation. The eigenvalues and are given in (38) and is determined by (35).

The 2-3 element of matrix (33) vanishes after additional 2-3 rotation by an angle :
which produces corrections of the next order in . With an additional 2-3 rotation the mixing matrix becomes
where
and the last 2-3 rotation is on the angle determined through . The expression on the RH of (41) is obtained by reducing the expression on the LH side to the standard form by permuting the correction matrix . According to (42), it is this matrix that leads to the modification of 2-3 mixing and CP phase in matter. From (42) one finds
that is, the combination is invariant under inclusion of matter effects. Furthermore, and up to corrections of the order . The results described here allowing understand the behavior of the mixing parameters in the region of the 1–3 resonance and above it (see Figure 1).

In Figure 2 we present dependence of the flavor content of the neutrino eigenstates on the potential. The energy level scheme, the dependence of the eigenvalues on matter density, is shown in Figure 3. The energy levels in matter do not depend on or , but they do depend on the 1–3 and 1-2 mixing.

Figure 2: The flavor contents of the eigenstates of the Hamiltonian in matter as functions of . The vertical width of the band is taken to be 1, then the vertical sizes of the colored parts give (red), and (green), (blue). The right and left panels correspond to neutrinos and antineutrinos, respectively. We take the best fit values of [21] with . Variations of change the relative and contents. The dashed red line shows a shift of border between and flavors for . The upper (lower) panel corresponds to normal (inverted) mass ordering.

Figure 3: The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter on . Note that we are plotting , which goes to for low . The left (right) panel corresponds to normal (inverted) mass ordering.

In the case of normal mass hierarchy, there are two resonances (level crossings) whose location is defined as the density (energy) at which the mixing in a given channel becomes maximal.(1)The H resonance, in the channel, is associated to the 1–3 mixing and large mass splitting. According to (31) at
(2)The L resonance at low densities is associated to the small mass splitting and 1-2 mixing. It appears in the channel, where and differ by small (at low densities) rotation given by an angle (see (31)). According to (37) the position of the L-resonance, , is given by , where is defined in (34). This leads to

For antineutrinos ( in Figure 3), the oscillation parameters in matter can be obtained from the neutrino parameters taking and . The mixing pattern and level scheme for neutrinos and antineutrinos are different both due to the possible fundamental violation of CP invariance and the sign of matter effect. Matter violates CP invariance and the origin of this violation stems from the fact that usual matter is CP asymmetric; in particular, there are electrons in the medium but no positrons.

In the case of normal mass hierarchy there are no antineutrino resonances (level crossings), and with the increase of density (energy) the eigenvalues have the following asymptotic limits:

3. Effects of Neutrino Propagation in Different Media

3.1. The Evolution Matrix

The evolution matrix, , is defined as the matrix which gives the wave function of the neutrino system at an arbitrary moment once it is known in the initial moment :
Inserting this expression in the evolution equation (12), we find that satisfies the same evolution equation as :
The elements of this matrix are the amplitudes of transitions: . The transition probability equals . The unitarity of the evolution matrix, , leads to the following relations between the amplitudes (matrix elements):
The first and the second equations express the fact that the total probability of transition of to everything is one, and the same holds for . The third and fourth equations are satisfied if
With these relations the evolution matrix can be parametrized as

The Hamiltonian for a system is T symmetric in vacuum as well as in medium with constant density. In medium with varying density the T symmetry is realized if the potential is symmetric. Under T transformations , and the diagonal elements do not change. Therefore according to (50) the T invariance implies that , or ; that is, the off-diagonal elements of the matrix are pure imaginary.

3.2. Neutrino Oscillations in Matter with Constant Density

In a medium with constant density and therefore constant potential the mixing is constant: . Consequently, the flavor composition of the eigenstates does not change and the eigenvalues of the full Hamiltonian are constant. The two-neutrino evolution equation in matter of constant density can be written in the matter eigenstate basis as
where . This system of equations splits and the integration is trivial, . The corresponding matrix is diagonal:
where is the half-oscillation phase in matter and a matrix proportional to the unit matrix has been subtracted from the Hamiltonian.

The matrix in the flavor basis is therefore
Then, for the transition probability, we can immediately deduce
where , with
being the oscillation length in matter. The dependence of on the neutrino energy is shown in Figure 4. For small energies, , the length . It then increases with energy and for small reaches the maximum at , that is, above the resonance energy. For the oscillation length converges to the refraction length .

Figure 4: Dependence of the oscillation length in matter in units of the refraction length on neutrino energy for two different mixing angles in vacuum.

A useful representation of the matrix for a layer with constant density follows from (54):
where is a vector containing the Pauli matrices and .

The dynamics of neutrino flavor evolution in uniform matter are the same as in vacuum, that is, it has a character of oscillations. However, the oscillation parameters (length and depth) differ from those in vacuum. They are now determined by the mixing and effective energy splitting in matter: , .

3.3. Neutrino Polarization Vectors and Graphic Representation

It is illuminating to consider the dynamics of transitions in different media using graphic representation [22–24]. Consider the two-flavor neutrino state, . The corresponding Hamiltonian can be written as
where , is the Hamiltonian vector , and is the oscillation length. The evolution equation then becomes
Let us define the polarization vector
In terms of the wave functions, the components of equal
The -component can be rewritten as ; therefore and from unitarity . Hence, determines the probabilities to find the neutrino in a given flavor state. The flavor evolution of the neutrino state corresponds to a motion of the polarization vector in the flavor space. The evolution equation for can be obtained by differentiating (60) with respect to time and inserting and from evolution equation (59). As a result, one finds that
If is identified with the strength of a magnetic field, the equation of motion (62) coincides with the equation of motion for the spin of electron in the magnetic field. According to this equation precesses around .

With an increase of the oscillation phase (see Figure 5) the vector moves on the surface of the cone having axis . The cone angle , the angle between and , depends both on the mixing angle and on the initial state, and, in general, on changes in process of evolution, for example, if the neutrino evolves through several layers of different density. If the initial state is , the angle equals in the initial moment.

Figure 5: Graphic representation of neutrino oscillations. Neutrino polarization vector precesses around the Hamiltonian vector (or the vector of eigenstates of the Hamiltonian). The angle between and is given by the cone angle , and the direction of axis of the cone is determined by the mixing angle in matter .

The components of the polarization vector are nothing but the elements of the density matrix . The evolution equation for can be obtained from (62)
The diagonal elements of the density matrix give the probabilities to find the neutrino in the corresponding flavor state.

3.4. Resonance Enhancement of Oscillations

Suppose a source produces flux of neutrinos in the flavor state with continuous energy spectrum. This flux then traverses a layer of length with constant density . At the end of this layer a detector measures the component of the flux, so that oscillation effect is given by the transition probability . In Figure 6 we show dependence of this probability on energy for thin and thick layers. The oscillatory curves are inscribed in the resonance envelope . The period of the oscillatory curve decreases with the length . At the resonance energy,
oscillations proceed with maximal depths. Oscillations are enhanced up to in the resonance range where (see Section 2.4). This effect was called the resonance enhancement of oscillations.

Figure 6: Resonance enhancement of neutrino oscillations in matter with constant density. Shown is the dependence of the transition probability on energy for for three different sizes of layers: , , and . The shaded area shows the resonance envelope: .

3.5. Three-Neutrino Oscillations in Matter with Constant Density

The oscillation probabilities in matter with constant density have the same form as oscillation probabilities in vacuum and the generalization of (53) is straightforward. In the basis of the eigenstates of the Hamiltonian the evolution matrix equals
and for the elements of the matrix in the flavor basis we obtain . Removing and using the unitarity of the mixing matrix in matter we have
In particular, for the amplitudes in matter involving only and , we obtain

3.6. Propagation in a Medium with Varying Density and the MSW Effect

3.6.1. Equation for the Instantaneous Eigenvalues and the Adiabaticity Condition

In nonuniform media, the density changes along neutrino trajectory: . Correspondingly, the Hamiltonian of system depends on time, , and therefore the mixing angle changes during neutrino propagation: . Furthermore, the eigenstates of the instantaneous Hamiltonian, and , are no longer the “eigenstates” of propagation. Indeed, inserting in the equation for the flavor states (c.f., (3)) we obtain the evolution equation for eigenstates :
where . The Hamiltonian for (68) is nondiagonal and, consequently, the transitions occur. The rate of these transitions is given by the speed with which the mixing angle changes with time. According to (68) [3, 25], determines the energy of transition and gives the energy gap between the levels.

The off-diagonal elements of the evolution equation (68) can be neglected if is much smaller than other energy scales in the system. The difference of the diagonal elements of the Hamiltonian is, in fact, the only other energy quantity and therefore the criterion for smallness of is
This inequality implies a slow enough change of density and is called the adiabaticity condition. Defining the adiabaticity parameter [22, 25] as
the adiabaticity condition can be written as .

For small mixing angle, the adiabaticity condition is most crucial in the resonance layer where the level splitting is small and the mixing angle changes rapidly. In the resonance point, it takes the physically transparent form [3]: , where is the oscillation length in resonance, and is the spatial width of the resonance layer. According to this condition at least one oscillation length should be obtained within the resonance layer.

In the case of large vacuum mixing, the point of maximal adiabaticity violation [26, 27] is shifted to density, , larger than the resonance density: . Here is the density at the border of resonance layer for maximal mixing. Outside the resonance and in the nonresonant channel, the adiabaticity condition has been considered in [28, 29].

3.7. Adiabatic Conversion and the MSW Effect

If the adiabaticity condition is fulfilled and can be neglected, the Hamiltonian for the eigenstates becomes diagonal. Consequently, the equations for the instantaneous eigenstates split as in the constant density case. The instantaneous eigenvalues evolve independently, but the flavor content of the eigenstates changes according to the change of mixing in matter. This is the essence of the adiabatic approximation; we neglect in evolution equation but do not neglect the dependence of on density. The solution can be obtained immediately as
in symmetric form. The only difference from the constant density case is that the eigenvalues now depend on time and therefore integration appears in the phase factors.

The evolution matrix in the flavor basis can be obtained by projecting back from the eigenstate basis to the flavor basis with the mixing matrices corresponding to initial and final densities:
From this procedure we find, for example, the probability of transition
If the initial and final densities coincide, as in the case of neutrinos crossing the Earth, we obtain the same formulas as in constant density case:
with the mixing angle taken at the borders (initial or final state). In particular, the survival probability equals .

Averaging over the phase, which means that the contributions from and add incoherently, gives
The mixing in the neutrino production point is determined by density in this point, , and the resonance density. Consequently, the picture of the conversion depends on how far from the resonance layer (in the density scale) a neutrino is produced. Strong transitions occur if the initial and final mixings differ substantially, which is realized when the initial density is much above the resonance density and the final one is below the resonance density and therefore neutrinos cross the resonance layer.

According to (73) the oscillation depth equals . Both the averaged probability (75) and the depth (73) are determined by the initial and final densities and do not depend on the density distribution along the neutrino trajectory. Essentially they are determined by the ratios in the initial and final moments. This is a manifestation of the universality of the adiabatic approximation result.

In contrast, the phase does depend on the density distribution and the period of oscillations (the latter is given by the oscillation length in matter). So, it is the phase that encodes information about the density distribution.

The probability depends on via the phase and also via the mixing angle . Two degrees of freedom are operative and dependence on time is an interplay of two effects: oscillations, associated with the phase , and the adiabatic conversion related to change of . Depending on initial condition , the relative importance of the two effects is different. If neutrinos are produced far above the resonance, , the initial mixing is strongly suppressed, . Consequently, the neutrino state, for example, , consists mainly of one eigenstate, , and furthermore, one-flavor dominates in . Since the admixture of the second eigenstate is very small, oscillations (interference effects) are strongly suppressed. Thus, here the nonoscillatory flavor transition occurs when the flavor of whole state (which nearly coincides with ) follows the density change. At zero density , and therefore the probability to find the electron neutrino (survival probability) equals [3]
The final probability, , is the feature of the nonoscillatory transition (as pure adiabatic conversion). Deviation from this value indicates the presence of oscillations; see (73).

If neutrinos are produced not too far from resonance, for example, at , the initial mixing is not suppressed. Although is the main component of the neutrino state, the second eigenstate, , has appreciable admixture; the flavor mixing in the neutrino eigenstates is significant, and the interference effect is not suppressed. Here we deal with the interplay of the adiabatic conversion and oscillations.

Production in the resonance is a special case; if , the averaged probability equals independently of the final mixing. This feature is important for determining the oscillation parameters. Strong transitions () occur when neutrinos cross resonance layer. These features are realized for solar neutrinos when propagating from their production region inside the Sun to the surface of the Sun. The adiabatic propagation occurs also in a single layer of the Earth (e.g., in the mantle).

3.8. Adiabaticity Violation

For most of applications the adiabaticity is either well satisfied (neutrinos in the Sun or supernovae), or maximally broken due to sharp (instantaneous) density change (neutrinos in the Earth, neutrinos crossing the shock wave fronts in supernova). In the former case the evolution is described by the adiabatic formulas. In the latter case description is also simple; one just needs to match the flavor conditions at the borders between layers, find the flavor state before the density jump, and then use it as an initial state for the evolution after the jump. The intermediate case of the adiabaticity breaking can be realized for neutrinos in the mantle of the Earth, for high energy neutrinos propagating in the Sun (neutrinos from annihilation of hypothetical WIMPs) or for sterile neutrinos with very small mixing.

If the density changes rapidly, is not negligible in (68) and the adiabaticity condition (70) is not satisfied. The transitions become noticeable and therefore the admixtures of the eigenstates in a given propagating state change. The matrix in the flavor basis is given by
where is the evolution matrix in the basis of instantaneous eigenstates. Then the transition probability equals
where is the probability of transitions and is an interference term
which depends on the oscillation phase. The averaged probability () equals [30]
If the initial density is much larger than the resonance density, then and . In this case the averaged probability can be rewritten as
Violation of adiabaticity weakens transitions if , thus leading to an increase of the survival probability. In the adiabatic case , , and therefore , so that (78) is reduced to (73).

In the graphic representation (Figure 5), the neutrino vector moves on the surface of the cone (phase change) and the axis of the cone rotates according to the density change. The cone angle changes as a result of violation of the adiabaticity).

There are different approaches to compute the flop probability . In the adiabatic regime the probability of transition between the eigenstates is exponentially suppressed with given in (70) [30, 31]. One can consider such a transition as penetration through a barrier of height by a system with the kinetic energy . This leads to the Landau-Zener probability
where [32]. In the case of weak adiabaticity violation, one can develop an adiabatic perturbation theory which gives the results as a series expansion in the adiabaticity parameter [33].

3.9. Theory of Small Matter Effects

3.9.1. Minimal Width Condition

If the vacuum mixing angle is small, there exists a lower limit on the amount of matter needed to induce significant flavor change due to matter effect. The amount of matter is characterized by the column density of electrons along the neutrino trajectory:
We can define as the column density for which the oscillation transition probability surpasses for the first time in the course of propagation. Then it is possible to show that [34]
for all density profiles. Furthermore, the minimum, , is realized for oscillations in a medium of constant density equal to the resonance density. The relation (84) is known as the minimal width condition. This condition originates from an interplay between matter effects and vacuum mixing. The acquired matter phase, , must be large. At the same time, since matter effects by themselves are flavor conserving, also vacuum mixing is required in order to induce flavor conversion. The smaller the vacuum mixing is, the larger the width that is required.

3.9.2. Vacuum Mimicking

Vacuum mimicking [35], which states that regardless of the matter density, the initial flavor evolution of neutrino state is similar to that of vacuum oscillations. Consequently for small baselines, , it is not possible to see matter effect and any such effect appearing in higher order of . Indeed, consider the evolution matrix
where denotes time ordering of the exponential. For small values of , it can be expanded as
If initial neutrino state has definite flavor, the amplitude of flavor transition is given by the off-diagonal element of which does not depend on matter potential. The matter contribution to is diagonal. Therefore the flavor transitions depend on the matter density only at higher order in . This result holds true as long as or when the phase of oscillation is small [36].

This can be seen explicitly in the case of medium with constant density where, expanding the oscillatory factor for small oscillation phase, we have the transition probability

Note that vacuum mimicking only occurs if the initial neutrino state is a flavor eigenstate [36]. If the initial neutrino is in a flavor-mixed state, for example, in a mass eigenstate, then matter will affect this state already at lowest order in . This situation is realized in several settings involving astrophysical neutrinos propagating through the Earth, for example, solar and supernova neutrinos, where the neutrinos arrive at the Earth as mass eigenstates. The mimicking is not valid if there are nonstandard flavor changing interactions, so that matter effect appears in the off-diagonal elements of the Hamiltonian.

3.9.3. Effects of Small Layers of Matter

If the minimal width condition is not satisfied, that is , the matter effect on result of evolution is small. This inequality can be written as which means that the oscillation phase is small. In this case the matter effect can be considered as small perturbation of the vacuum oscillation result even if the MSW resonance condition is satisfied.

The reasons for the smallness of the matter effect are different depending on the energy interval. Consider a layer of constant density with the length . There are three possibilities.(i) ( is the resonance density)—nearly vacuum oscillations in low density medium take place. Matter effect gives small corrections to the oscillation depth and length which are characterized by , here .(ii)—modification of oscillation parameters is strong; however . Consequently, . Oscillations are undeveloped due to smallness of phase.(iii)—matter suppresses oscillation depth by a factor . Since the oscillation length equals , one obtains . Hence in this case the distance is very small and oscillation effect in the layer has double suppression.

3.10. Propagation in Multilayer Medium

3.10.1. Parametric Effects in the Neutrino Oscillations

The strong transitions discussed in the previous sections require the existence of large effective mixing, either in the entire medium (constant density) or at least in a layer (adiabatic conversion). There is a way to get strong transition without large vacuum or matter mixings. This can be realized with periodically or quasiperiodically changing density [24, 37] when the conditions of parametric resonance are satisfied. Although the flavor conversion in a layer which corresponds to one period is small, strong transitions can build up over several periods. For large mixing even a small number of periods are enough to obtain strong flavor transitions.

The usual condition of parametric resonance is that the period of density change is an integer times the effective oscillation length [38]:
or . Such an enhancement has been considered first for modulation of the profile by sine function [39]. This may have some applications for intense neutrino fluxes when neutrino-neutrino interactions become important.

The solvable case, which has simple physical interpretation, is provided by the castle wall profile, for which the period is divided into the two parts and () with the densities and , respectively ( and, in general, ). Thus, the medium consists of alternating layers with two different densities [37, 40–45].

For the “castle wall” profile, the simplest realization of the parametric resonance condition is reduced to equality of the oscillation phases acquired by neutrinos over the two parts of the periods [41]:

The enhancement of transition depends on the number of periods and on the amplitude of perturbation, which determines the swing angle (the difference of the mixing angles in the two layers, ). For small a large transition probability can be achieved after many periods. For large “swing” angle, even a small number of periods are sufficient.

3.10.2. Parametric Enhancement: General Consideration

In general the condition (89) is not necessary for the enhancement or even for maximal enhancement. First, consider the oscillation effect over one period. The corresponding evolution matrix is given by the product
where is the evolution in layer given by (57). For brevity we will write it as , where , , and is the half phase acquired in layer :
Here is the mixing angle in layer .

Insertion of from (57) into (90) gives [37]
where
Explicitly, and . Using unitarity of , which gives , one can parametrize and with a new phase as and . Then the evolution matrix can be written in the form , where . Consequently, the evolution matrix after periods equals
It is simply accounted for by an increase of the phase: . This is the consequence of the fact that the evolution matrices over all periods are equal and therefore commute. If the evolution ends at some instant which does not coincide with the end of a full period, that is, , then .

The transition probability computed with (94) is
It has the form of the usual oscillation probability with phase and depth . The oscillations described by (95) are called the parametric oscillations. Under condition
which is called the parametric resonance condition, the depth of oscillations (95) becomes 1 and the transition probability is maximal when , where is an integer. There are different realizations of the condition (96) which imply certain correlations among the mixing angles and phases. The simplest one, , coincides with (89).

3.10.3. Parametric Enhancement in Three Layers

For small number of layers an enhancement of flavor transition can occur due to certain relations between the phases and mixing angles in different layers. This in turn imposes certain conditions on the parameters of the layers: their densities and widths. The conditions are similar to the parametric resonance condition and this enhancement is called the parametric enhancement of flavor transitions. These conditions can be satisfied for certain energies and baselines for neutrinos propagating in the Earth.

Consider conditions for maximal enhancement of oscillations for a different number of layers. It is possible to show [46] that they are generalizations of the conditions in one layer which require that (i) the depth of oscillations is 1 (we call it the amplitude condition) and (ii) the oscillation phase is —the phase condition.

Consider first the case of one layer with (in general) varying density (it can correspond to the mantle crossing trajectories in the Earth). The resonance condition for constant density case, , can be written according to (22) and (51) as , that is, , or equivalently, , where the superscript indicates the number of layers. This generalization goes beyond the original MSW resonance condition (even for constant density). The phase condition can be rewritten in terms of the elements of the evolution matrix (c.f., (54)) as . The absolute maximum of the transition probability occurs when these conditions are satisfied simultaneously, that is, when .

The parametric resonance condition (96) can be generalized to the case of nonconstant densities in the layers although the generalization is not unique. Indeed, according to (92) the condition can be written in terms of the elements of the evolution matrix for the two layers as the equality of the diagonal elements . Let us find the conditions for extrema for density profiles consisting of two layers. We have , where , and for each layer have been defined in (51). The sum of the two complex numbers in the transition amplitude has the largest possible result if they have the same phase: , which can also be rewritten as
This condition is called the collinearity condition [46]. It is an extremum condition for the two-layer transition probability under the constraint of fixed transition probabilities in the individual layers. In other words, if the absolute values of the transition amplitudes are fixed while their arguments are allowed to vary, then the transition probability reaches an extremum when these arguments satisfy (97).

The conditions for maximal transition probability for three layers can be found in the following way. The 1-2 elements of the evolution matrix equal
In the case of neutrino oscillations in the Earth, the third layer is just the second mantle layer, and its density profile is the reverse of that of the first layer. The evolution matrix for the third layer is therefore the transpose of that for the first one [47]; that is, , , and the expression for can be written as
Note that is pure imaginary because the core density profile is symmetric. Therefore the amplitude in (99) is also pure imaginary, as it must be because the overall density profile of the Earth is symmetric as well. If the collinearity condition for two layers (97) is satisfied, then not only the full amplitude , but also each of the four terms on the right-hand side of (99) is pure imaginary. If the collinearity condition is satisfied for two layers, then it is automatically satisfied for three layers. This is a consequence of the facts that the density profile of the third layer is the reverse of that of the first layer and that the second layer has a symmetric profile. The conditions described here allow reproducing very precisely all the main structures of the oscillograms of the Earth (see Section 4.1).

3.11. Oscillations of High Energy Neutrinos

At high energies or in high density medium when , we can use as a small parameter and develop a perturbation theory using its smallness. However, in most situations of interest, the neutrino path length in matter is so large that . Therefore the vacuum part of the Hamiltonian cannot be considered as a small perturbation in itself and the effect of on the neutrino energy level splitting should be taken into account. For this reason we split the Hamiltonian as with
where is the oscillation frequency (20) and . The ratio of the second and the first terms in the Hamiltonian (100) is given by the mixing angle in matter : . Therefore for the term can be considered as a perturbation. Furthermore, , so the diagonal terms in can be neglected in the lowest approximation.

The solution for matrix can be found in the form , where is the solution of the evolution equation with replaced by (see (71)). The matrix then satisfies the equation
where is the perturbation Hamiltonian in the “interaction” representation. Equation (101) can be solved by iterations: , which leads to the standard perturbation series for the matrix. For neutrino propagation between and we have, to the lowest non-trivial order,
The transition probability is given by

For density profiles that are symmetric with respect to the center of the neutrino trajectory, , (103) gives
where is the distance from the midpoint of the trajectory and is the phase acquired between this midpoint and the point . The transition probability decreases with the increase of neutrino energy essentially as . The accuracy of (103) also improves with energy as .

Inside the Earth, the accuracy of the analytic formula is extremely good already for GeV. When neutrinos do not cross the Earth’s core () and so experience a slowly changing potential , the accuracy of the approximation (103) is very good even in the MSW resonance region – GeV.

The above formalism applies in the low energy case as well, with only minor modifications: the sign of in (100) has to be flipped, and correspondingly one has to replace in the definition of . The expressions for the transition probability in (103) and (104) remain unchanged.

3.12. Effects of Small Density Perturbations

Let us consider perturbation around smooth profile for which exact solution is known. The simplest possibility that has implications for the Earth matter profile is the constant density with additional perturbation: . Correspondingly, the Hamiltonian of the system can be written as the sum of two terms:
where
Here, is the mixing angle in matter and is half of the energy splitting (half-frequency) in matter, both with the average potential . We will denote by the evolution matrix of the system for the constant density case . The expression for is given in (54) with and , .

The solution of the evolution equation with Hamiltonian (105) [46] is of the form
where satisfies . Inserting (107) into the evolution equation, one finds the following equation for to the first order in and :
where . The first term in (108) does not contribute to since , and (108) can be immediately integrated:
Introducing the distance from the midpoint of the neutrino trajectory , one obtains from (109)
where . In these integrals, and . The integral vanishes if the perturbation is symmetric with respect to the midpoint of the trajectory. Analogously, vanishes if is antisymmetric. The expression for defined in (107) is equivalent to (13)–(16) obtained in [48] in the context of solar neutrino oscillations.

For practical purposes it is useful to have an expression for which is exactly unitary regardless of the size of the perturbation. For this we rewrite (110) as follows:
where and . Thus, and we replace it by
Here both and are unitary matrices, and due to their specific form the combination on the right-hand side of (112) is exactly unitary.

For a symmetric density profile with respect to the midpoint of the trajectory, the term is absent. From (54), (110), and (112) we immediately get the transition probability